14:332:423 Computer
and Communication Networks

Note: A more complete introduction along with packet-by-packet Fair
Queuing and WFQ is available in Chapter 5
of my Computer Networking book [downloadable PDF document].

We often run into the problem of dividing a scarce resource among a
set of users, each of whom has an equal right to the resource, but
some of whom intrinsically demand fewer resources than others. How,
then, should we divide the resource? A sharing technique widely used
in practice is called max-min fair share. Intuitively, a
fair share allocates a user with a "small" demand what it wants, and
evenly distributes unused resources to the "big" users. Formally, we
define max-min fair share allocation to be as follows:

Resources are allocated in order of increasing demand

No
source gets a resource share larger than its demand

Sources
with unsatisfied demands get an equal share of the resource

This formal definition corresponds to the following operational
definition. Consider a set of sources 1, ..., n that have
resource demands x1, x2, ...,
xn. Without loss of generality, order the source
demands so that x1 <= x2 <= ... <=
xn. Let the server have capacity C. Then,
we initially give C/n of the resource to the source with the
smallest demand, x1. This may be more than what
source 1 wants, perhaps, so we can continue the process. The process
ends when each source gets no more than what it asks for, and, if its
demand was not satisfied, no less than what any other source with a
higher index got. We call such an allocation a max-min fair
allocation, because it maximizes the minimum share of a source whose
demand is not fully satisfied. EXAMPLE 1

Compute the max-min fair allocation for a set of four sources with
demands 2, 2.6, 4, 5 when the resource has a capacity of 10.

Solution: We compute the fair share in several rounds of
computation. In the first round, we tentatively divide the resource
into four portions of size 2.5. Since this is larger than source 1's
demand, this leaves 0.5 left over for the remaining three sources,
which we divide evenly among the rest, giving them 2.66... each. This
is larger than what source 2 wants, so we have an excess of 0.066...,
which we divide evenly among the remaining two sources, giving them
2.5 + 0.66... + 0.033... = 2.7 each. Thus, the fair allocation is:
source 1 gets 2, source 2 gets 2.6, sources 3 and 4 get 2.7 each.

Thus far, we have assumed that all sources have the same right to the
resources. Sometimes, we may want to give some sources a bigger share
than others. In particular, we may want to associate weights
w1, w2, ..., wn with sources 1,
2, ..., n, which reflect their relative resource share. We
extend the concept of max-min fair share to include such weights by
defining the max-min weighted fair share allocation as
follows:

Resources are allocated in order of increasing demand, normalized
by the weight

No source gets a resource share larger than its demand

Sources with unsatisfied demands get rerource shares in
proportion to their weights

The following example shows how to achieve this in practice.
EXAMPLE 2

Compute the max-min fair allocation for a set of four sources with
demands 4, 2, 10, 4 and weights 2.5, 4, 0.5, 1 when the resource has a
capacity of 16.

Solution:The first step is to normalize the weights so
that the smallest weight is 1. This gives us the set of weights as 5,
8, 1, 2. We can now pretend that the number of sources, instead of
being 4, is 5 + 8 + 1 + 2 = 16. We therefore divide the resource into
16 shares. In each round of resource allocation, we give a source a
share proportional to its weight. Thus, in the first round, we
compute C/n as 16/16 = 1. In this round, the sources receive
5, 8, 1, 2 units of the resource, respectively. Source 1 gets 5, and
only needs 4, so we have 1 unit extra. Similarly, source 2 has 6
units extra. Sources 3 and 4 are backlogged, since their share is
less than their demand. We now have 7 units of resources which have
to be distributed to sources 3 and 4. Their weights are 1 and 2, and
the smallest weight is 1, so there is no need to normalize weights.
We give source 3 an additional 7 × 2/3 units (since its weight is
1), and source 4 an additional 7 × 2/3 units (since its weight is
2). This increases source 4's share to 2 + 7 × 2/3 = 6.666
units, which is more than it needs. So we give the excess 2.666 units
to source 3, which finally gets 1 + 7/3 + 2.666 = 6 units. The final
shares are, therefore, 4, 2, 6, 4. This is the max-min weighted fair
share allocation.